In this article we investigate the modeling and computational aspects of the product-of-exponentials (POE) formula for robot kinematics. While its connections with Lie groups and Lie algebras give the POE equations mathematical appeal, little is known regarding its usefulness for control and other applications. We show that the POE formula admits a simple global interpretation of an open kinematic chain and possesses several useful device-independent features absent in the Denavit-Hartenberg (DH) representations. Methods for efficiently computing the forward kinematics and Jacobian using these equations are presented. In particular, the computational requirements for evaluating the Jacobian from the POE formula are compared to those of the recursive methods surveyed in Orin and Schrader (1984). >

Computational aspects of the product-of-exponentials formula for robot kinematics

This paper sets forth an approach to robotic kinematics that brings together the use of matrix exponentiation and dual numbers. Working from first principles, it is shown that a screw displacement can be represented by the exponential of a dual skew-symmetric matrix; a product of two such matri ces will suffice to specify the relationship between a pair of link coordinate frames. An example problem, dealing with the kinematics of the Rhino XR-2, illustrates how the approach immediately identifies those groups of joint variables that tend to cluster together.

On the Use of Dual- Matrix Exponentials in Robotic Kinematics

A number of conventional methods for orientation representation are reviewed and discussed. An analysis based on Lie algebra for exploring possible definitions of three-dimensional (3D) orientation vectors and for unifying representations of position and orientation is presented. The associated computation methods are developed and verified by application to the six-axis Puma 560 robot. >

An exploration of orientation representation by Lie algebra for robotic applications

An investigation is conducted of the computational aspects of the product of exponentials (POE) formula, and an application to resolved-rate motion control of a six-degree-of-freedom manipulator based on a table lookup scheme is presented. It is shown that the computational requirements for evaluating the Jacobian are comparable to the more standard recursive methods discussed in the work of D. Orin and W. Schrader (1984). In addition to having intuitive geometric appeal, the POE formula adds some measure of device-independence to the kinematic equations of a manipulator. >

Motion control using the product-of-exponentials kinematic equations

Robot motion is described by a set of positions and orientations of the robot's end-effector relative to some base coordinate system. Different methods to describe this motion are available and most of them are presented in this paper. Methods for position vector, free vector, and line vector transformation are presented and compared, and the advantages and disadvantages, the simplicity, and the number of arithmetic operations required of the different methods are addressed. An example of the application of these methods to a six degree-of-freedom robot is given.

On rotations and translations with application to robot manipulators

The exponential matrix representation in describing rotations has a geometric appeal in studying the kinematics of mechanisms and in characterizing their singularities. Here, we show its potential computational appeal using the classical Cayley-Hamilton theorem. In so doing, a connection between the geometric and computational aspects is clearly underlined. Then for the purpose of a demonstration, we perform explicit calculations to obtain the kinematics relations of the Stanford manipulator.

https://ieeexplore.ieee.org/iel5/9/24241/01104277.pdf

On the computational aspect of the matrix exponentials and their use in robot kinematics

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On the computational aspect of the matrix exponentials and their use in robot kinematics